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Questions tagged [groupoids]

A groupoid (in the sense of category theory) is a small category in which every morphism is an isomorphism. Groupoids arise throughout mathematics, e.g. in guise of fundamental groupoids in the theory of covering spaces, holonomy groupoids in the theory of foliations or Lie groupoids in mathematical physics. For groupoids in the sense of universal algebra, i.e., a set with a binary operation, please use the (magma) tag.

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Inversion on an internal groupoid

Let $\mathcal{C}$ be a category with pullbacks, with $\mathscr{G}=({\bf Ob}_\mathscr{G},{\bf Hom}_\mathscr{G},cod,dom,{\bf 1}, \circ_{\mathscr{G}},-^{-1})$ an internal groupoid in $\mathcal{C}$. I'm ...
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Faithful representation of $C_c(X \times X)$.

Let $X$ be a smooth manifold. $X \times X$ is product manifold. $\mu$ is a Borel measure on $X$. There are two aims (i) Associate a $C^*$ norm to $C_c(X\times X)$, making it a $C^*$ algebra. (ii)...
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Proving a that $C_c(TM)$ injects into $C_0(T^*M)$ through Fourier Transform

Let $M$ be a smooth manifold. Equip $TM$ with an Riemannian structure. We let $f \in C_c(TM)$ and define a homorphism into $C_0(T^*M)$ by $$ (x,w) \in T^*M, \hat{f}(x,w) = (2\pi)^{-\frac{n}{2}} \...
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Groupoid Theory I, Higson's notes

This is Example 7.22, page 100. We first consider $G=TM$ a smooth groupoid with base space $M$. The source and range maps are the same projection maps $s,r:TM \rightarrow M, X_m \mapsto m$. Now I ...
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Milnor construction and deloopings

To construct a classifying space (and universal bundle) of a topological group $G$ one can use the well-known Milnor construction based on the infinite join of $G$. On the other hand one can (at ...
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What is a topological groupoid?

I'm reading section 2.7 (Fundamental Group of the Circle) of the book Algebraic Topology by Tom Dieck. The section mentions the term "topological groupoid" but I cannot find the definition in previous ...
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Is the 2-category of groupoids a topos?

I have no justification for this, but I am wondering if the 2-category of groupoids is a topos.
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Object of a Category $C$ acts as Functor

I have a question about a notation used in following paper: https://etale.site/writing/stax-seminar-talk.pdf (see page 4): We take a category $C$ and consider a pair $(X_0,X_1)$ of two objects $X_1, ...
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Categories Fibered in Groupoids and Yoneda

My question refers to an article of Aaron Mazel-Gee about fibered categories in grupoids where the author introduces in a quite strange way a new category $\{X_0/X_1\}$ providing a functor $p: \{X_0/...
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Space of Riemannian metrics as a topological groupoid

I'm reading these notes on groupoids and I'm struggling with example 1.4. I recall the relevant definitions below. Definition: A groupoid $\mathcal{G}$ is a small category in which every arrow is ...
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Fibered Categories in Groupoids

I'm reading an article of Aaron Mazel-Gee about Fibered categories in grupoids and there is an example which I don't understand. Here the full article: https://etale.site/writing/stax-seminar-talk.pdf ...
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Can nonisomorphic groupoids have homotopy equivalent classifying spaces?

We know that two discrete groups having the same classifying space up to homotopy are isomorphic. One can just take fundamental groups and conclude. The situation with topological groups is subtler. ...
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Right (bi)adjoint of the inclusion of $\mathbf{Grpd}$ in $\mathbf{Cat}$

Let $\mathbf{Grpd}$ and $\mathbf{Cat}$ be respectively the 2-categories of small groupoids and of small categories. At the 1-categorical level, the inclusion $\mathbf{Grpd}\rightarrow\mathbf{Cat}$ has ...
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what is an ∞-group?

I was reading on nLab and I found the term infinity group. The definition is awfully abstract: An ∞-group is a group object in ∞Grpd. Equivalently (by the delooping hypothesis) it is a pointed ...
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Is every topological groupoid equivalent to a disjoint union of topological groups?

It's a fact that any groupoid is equivalent to a disjoint union of (deloopings of) groups. See, e.g. Proposition 4.3 of https://ncatlab.org/nlab/show/groupoid#PropertiesEquivalencesOfGroupoids. Does ...
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The Whitehead product and $\pi_{\leq 3} S^2$

Why does "the non-vanishing of the Whitehead bracket" imply that the fundamental 3-groupoid $\pi_{\leq 3} S^2$ of the two-shere cannot be strictified (as claimed here)?
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The $\mathbb{C}$ vector space monad on the 2-Category of Groupoids

In this post, I am asking about the existence of something called the "Vector Space Monad" on the 2-Category of groupoids (Grpd). In a comment, it was pointed out that the monad should exist due to ...
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Is there a simplicial set classifying subobjects of groupoids?

A $1$-groupoid can be thought of as a Kan complex in the usual way. Is there a simplicial set $\Omega$ such that the contravariant functors $\text{Sub}_{\mathbf{Gpd}}(-)$ and $\text{Hom}_{\mathbf{sSet}...
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Can we see FdHlb as a 2Category of groupoids?

Can we see a finite dimensional Hilbert space, $H$ as a groupoid if we include the unitary endomorphisms of $H$? It would be like a category with a single object and just isos. If so, can we take a ...
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Equivalence of Categories between the Fundamental Group and Groupoid

Let $X$ be a path connected space, and let $x \in X$. Then we have that $\pi_1 (X,x)$ is a full subcategory of $\Pi(X)$. So, the inclusion functor $J: \pi_1(X,x) \to \Pi(X)$ is an equivalence of ...
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Definition of $\pi_0 p^{-1}(u)$

In Ronnie Brown's Topology and groupoid, pg 263, 7.2.1 If $p:E\to B$ is a fibration of groupoids, there is an assignment, $$b_\#:\pi_0 p^{-1}[u]\to\pi_0 p^{-1}[v]$$ I did not see anywhere where ...
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Adjoint for pullback of sheaves on a topological groupoid

Consider a topological groupoid $G\xrightarrow{s} X$, with the arrow representing the domain map that associates to each morphism its domain. This induces a pullback functor $s^*:Sh(X)\to Sh(G)$. I ...
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“Eilenberg-MacLane property” for the classifying space of a groupoid

Given a groupoid $G$, its classifying space is defined as the standard geometric realisation of the nerve. My question is: since the classifying space of a group is the only space up to homotopy that ...
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All sheaves in a Grothendieck topos are generated by subobjects of powers of a suitable sheaf

Consider a topos $\mathcal E$. Butz and Moerdijk, in Representing topoi by topological groupoids, par. 2, say that one can find an object $S\in \mathcal E$ such that the subobjects of its powers (i.e.,...
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Morita-equivalence of groupoids and classifying topoi: correct definition

The first comment to this post points out that, given two (topological or localic) groupoids, they can be non-Morita-equivalent evenif their classifying topoi (topoi of equivariant sheaves) are ...
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Understanding the monadicity of groupoids over splittings

In the paper The shift functor and the comprehensive factorization for internal groupoids by Bourn, the author proves that for a fixed finitely complete category, the category of internal groupoids is ...
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Higher homotopy groups in terms of the fundamental groupoid

Let $X$ be a topological space. Then we can construct the following structure. Let an $n$-morphism be a map $I^n\to X$. We can view $n+1$ morphisms exactly as homotopies between $n$-morphisms. Let $f,...
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The étale topos of a scheme is the classifying topos of…?

By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. Butz. and I. Moerdijk. Representing topoi by topological ...
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Are there examples of unital and nuclear $C^*$-algebras satisfying the UCT that are not groupoid algebras of an amenable etale groupoid?

Jean Louis Tu showed that the (maximal) groupoid $C^*$-algebra of a groupoid satisfying the Haagerup property (which includes all amenable groupoids) will satisfy the UCT. I am curious if there are ...
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Foliations and groupoids in algebraic geometry

I am currently studying the theory of foliations and groupoids from a differentiable viewpoint, in particular Haefliger spaces. [See Segal, Classifying spaces related to foliations, and Moerdijk, ...
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Subsets of groups containing identity and inverses

Let $G$ be a (finite) group, containing a subset $H$. We suppose that $H$ contains the identity and that it is closed under taking inverses. What are some of the algebraic properties of subgroups ...
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What is the tangent Lie algebroid to a Lie groupoid?

How do you define the tangent Lie algebroid to a Lie groupoid? In this online note Lie Algebroids, Lie Groupoids and Poisson Geometry by Sébastien Racanière, it states that if $t\colon G_1\to G_0$ is ...
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Equivalence Between Covering Groupoids and Covering Spaces

On page 390 of Topology and Groupoids he discusses the equivalence of the category of Covering Groupoids and the category of Covering Spaces. Could someone give me some examples on how it would be ...
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1answer
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Fibration of Groupoids

To show this is a fibration, do I just show that given $p: x \to y$ in $\text{ob}(E)$, and a map $b: F(p) = y \to y'$ then naturally the map $\Gamma: p \to p'$ has image $F(\Gamma): y \to y'$ so that $...
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1answer
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Inversion map of a Lie groupoid is a diffeomorphism

The inversion map $Inv:\mathcal G\to \mathcal G$ of a Lie groupoid $\mathcal G$ is given by $Inv(g)=g^{-1}$. And I want to show this inversion map is a diffeomorphism. Any suggestions will be ...
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Does it make sense to examine properties of $(G,\circ)$ if we, in the meantime prove that $(G,\circ)$ is not a groupoid?

If, for example, we are given a set $G$ and an operation $\circ$ and we have to examine properties of that operation on that set (closeness, associativity, commutativity, exsistance of neutral, ...
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Does the existence of only right neutral mean that there can only exist right inverse?

Let $(G,\circ $) be groupoid. If there exists only right neutral, does that mean that there can exist only right inverse? To put it in another way, does the existence of only right neutral mean that ...
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Is the Subcategory of Infinite Sets a Groupoid?

The following exercise is given in the text "Algebra Chapter 0" by Aluffi: Construct a category of infinite sets and explain how it may be viewed as a full subcategory of Set. Let $\infty\text{-...
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“Finite flat” vs. “finite locally free” groupoids

Let $(\{X_{0},X_{1}\},\{d_{0},d_{1} : X_{1} \to X_{0}\})$ be a groupoid in schemes. In SGA 3, Exp. V, section 5 a), the following is claimed, where I think it is being assumed that $d_{0}$ and $d_{1}$ ...
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1answer
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Can a groupoid with at least two objects have a group structure(in a pathological way, if necessary)?

A groupoid is defined to be a (small)category where every morphism is an isomorphism. As we know every group can be viewed as as groupoid with a single object. I wonder if the converse of this is also ...
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tensor product and loop space multiplication are homotopic on $\mathbb{P}^\infty$

$\newcommand{\C}{\mathbb{C}}$ $\newcommand{\P}{\mathbb{P}}$ Let the isomorphism $\C^n \otimes \C^m \to \C^{nm}$ be given by the diagonal ordering $e_0\otimes f_0,e_1 \otimes f_0, e_0 \otimes f_1, e_2 \...
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Gleason Yamabe for groupoids

A colleague of mine seems convinced that there is a Gleason-Yamabe type theorem for locally compact groupoids. Does anyone know if this is true? If so, any references would be most appreciated.
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Why is a groupoid required to be a small category?

I'm learning category theory and every definition of a groupoid has required that a groupoid be a small category (or some equivalent requirement) but I don't really know why. So what is the reason ...
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Is the category of groupoids a Lawvere thory?

By which I mean a category of models for a Lawvere theory. I have not seen this anywhere, so I wonder if something goes wrong with this category.
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Easily visualizable examples of Lie groupoids?

The goal is to understand the Poisson cohomology of Poisson manifolds, which according to the nCatLab, is "just" the Lie algebroid cohomology of the corresponding Poisson Lie algebroid. Chasing down ...
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Representations of groupoid algebras

In reading through Khalkhali's Noncommutative Geometry text, I came across something I don't understand. Let $\mathfrak{G}$ be a discrete groupoid, and for each $x\in Obj(\frak{G})$, define the *-...
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1answer
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The difference between weak Kan complexes and Kan complexes

Let $X$ be a simplicial set and let $\wedge_i^2$ be the $i-th$ horn of the simplicial set $\Delta^2$. The Kan condition is the horn filling condition for $i=0,1,2$ and the weak Kan condition is the ...
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Coseparating family in the category of groupoids

Does the category of (small) groupoids admit a small coseparating/cogenerating set of objects? I suppose $\mathbf{Cat}$ doesn't, but so far I have no clue about $\mathbf{Grpd}$.
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Paracompact iff deformation retract of classifying space of every open cover?

This MO answer and its comments, suggest a cool characterization of paracompactness I have never seen before. For a space $B$ let $(U_i)$ be an open cover. Form the groupoid $\amalg_iU_i\times_B \...
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Adjunction between Kan complexes and Groupoids

We have the fundamental groupoid functor $\Pi: \operatorname{KanCompl} \to \operatorname{Grpd}$ and the nerve functor $N:\operatorname{Grp}\to \operatorname{KanCompl}$ and I am trying to see that ...